The Stacks project

Remark 38.24.4. Tracing the proof of Lemma 38.24.3 to its origins we find a long and winding road. But if we assume that

1. $f$ is of finite type,

2. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module,

3. $E = X_ s$, and

4. $S$ is the spectrum of a Noetherian complete local ring.

then there is a proof relying completely on more elementary algebra as follows: first we reduce to the case where $X$ is affine by taking a finite affine open cover. In this case $Z$ exists by More on Algebra, Lemma 15.20.3. The key step in this proof is constructing the closed subscheme $Z$ step by step inside the truncations $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}/\mathfrak m_ s^ n)$. This relies on the fact that flattening stratifications always exist when the base is Artinian, and the fact that $\mathcal{O}_{S, s} = \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{S, s}/\mathfrak m_ s^ n$.